**1. Introduction**

The main characteristic of nanofluid is the significant enhancement of the thermal properties of the base fluid. The term nanofluid comes back to a pioneering experimental research by Choi [1] in which a conclusion had been reached that the thermal conductivity of a base fluid is enhanced up to two times by adding a small amount of nanoparticles. In addition, some authors [2,3] found that the dispersion of a small amount of copper nanoparticles led to 40% of the thermal conductivity of the fluid, while adding a small amount of carbon nanotubes in ethylene glycol or oil led to 50%. Aly and Ebaid [4] considered five metallic and nonmetallic nanoparticles in a base of water, where an effective approach was introduced to derive the exact solution. One of the important results in the

field of nanofluid flow has been presented by Majumder [5], in which it was experimentally proven that nanofluidic flow exhibits partial slip against the solid surface, which can be characterized by the so-called slip length. Accordingly, the authors in [6] discussed the effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet at constant wall temperature. Furthermore, the no-slip condition is no longer valid for fluid flows at the micro- and nanoscale and, instead, a certain degree of tangential slip must be allowed [7,8]. Very recently, Sharma and Ishak [9] studied the second-order velocity slip effect on the boundary layer flow of Cu-water-based nanofluid with heat transfer over a stretching sheet. Their numerical results were based on the finite element method (FEM). A model for isothermal homogeneous-heterogeneous reactions in boundary layer flow of viscous fluid past a flat plate was studied by Merkin [10]. He presented the homogeneous reaction by cubic autocatalysis and the heterogeneous reaction by a first-order process and showed that the surface reaction is the dominant mechanism near the leading edge of the plate. Chaudhary and Merkin [11] studied the homogenous-heterogeneous reactions in boundary layer flow of viscous fluid. They found the numerical solution near the leading edge of a flat plate. Bachok et al. [12] focused on the stagnation-point flow towards a stretching sheet with homogeneous-heterogeneous reactions effects. Effects of homogeneous-heterogeneous reactions on the flow of viscoelastic fluid towards a stretching sheet were investigated by Khan and Pop [13]. Kameswaran et al. [14] extended the work of [13] for nanofluid over a porous stretching sheet. In general, porous medium is used for transport and storage of energy. Analysis of flow through a porous medium has become the core of several scientific and engineering applications. These applications include the utilization of geothermal energy, the migration of moisture in fibrous insulation, food processing, casting and welding in manufacturing processes, the dispersion of chemical contaminants in different industrial processes, the design of nuclear reactors, chemical catalytic reactors, compact heat exchangers, solar power, and many others. Further, the use of micro/nano electromechanical systems (MEMS/NEMS) has been increased in many industries. Such systems have association with velocity slip [15–19]. Very recently, Hayat et al. [20] studied the MHD flow of nanofluid with homogeneous-heterogeneous reactions of two chemical species and velocity slip. In this field of research, some pioneer works were introduced in [21–24] in which several non-Newtonian models have been analyzed. In [21], a novel radiation MHD activation energy Carreau and nanofluid effects of thermal energy systems have been investigated. The combined electrical MHD Ohmic dissipation forced and free convection of an incompressible Maxwell fluid on a stagnation point heat and mass transfer energy conversion problem have been studied in [22]. In addition, an applied thermal system for heat and mass transfer and energy managemen<sup>t</sup> problem of hydromagnetic flow with magnetic and viscous dissipation effects of micropolar nanofluids towards a stretching sheet has been investigated by [23]. Moreover, the effect of the slip boundary condition on the stagnation electrical MHD nanofluid mixed convection on a stretching sheet was introduced in [24].

The objective of this work is to extend the model investigated by Hayat et al. [20] by considering the second-order slip velocity. Therefore, the extended model is given as

$$f'''(\eta) = \left(\lambda + (1 - \phi)^{2.5}M\right)f'(\eta) - \phi\_1\left(f(\eta)f''(\eta) - \left(f'(\eta)\right)^2\right),\tag{1}$$

$$\frac{1}{Sc}\mathbf{g''}(\eta) = \mathbf{Kg}(\eta)(h(\eta))^2 - f(\eta)\mathbf{g'}(\eta),\tag{2}$$

$$\frac{\delta}{Sc}h''(\eta) = -Kg(\eta)(h(\eta))^2 - f(\eta)h'(\eta),\tag{3}$$

subject to

$$f(0) = 0, f'(0) = 1 + \gamma f''(0) + \mu f'''(0), f'(\infty) = 0,\tag{4}$$

$$\mathcal{g}'(0) = K\_{\mathfrak{s}} \mathcal{g}(0), \mathcal{g}(\infty) = 1,\tag{5}$$

$$
\delta h'(0) = -K\_s g(0), \\
h(\infty) = 0,\tag{6}
$$

where

$$\phi\_1 = (1 - \phi)^{2.5} \left( 1 - \phi + \phi \frac{\left(\rho \mathbf{C}\_p\right)\_s}{\left(\rho \mathbf{C}\_p\right)\_f} \right), \tag{7}$$

and *φ* is the solid volume fraction of the nanoparticles, *λ* is the porosity parameter, *M* is the Hartman number, *Sc* is the Schmidt parameter, *K* is the measure of the strength of the homogeneous reaction, *Ks* is the measure of the strength of the heterogeneous reaction, *δ* is the ratio of the diffusion coefficient, *ρs* and *ρf* are respectively the densities of nanoparticles and base fluid, *γ* and *μ* are respectively the first and the second velocity slip parameters, and *f* (*η*), *g*(*η*) and *h*(*η*) are respectively the fluid velocity and the concentrations of the two chemical species.

Following [20], the parameter *δ* can be taken as unity especially when the diffusion coefficients of two chemical species are the same. In this case, we have [20]

$$h(\eta) + \gcd(\eta) = 1,\tag{8}$$

and hence Equations (2) and (3) reduce to

$$\frac{1}{Sc}\mathbf{g''}(\eta) = \mathbf{Kg}(\eta)(1 - \mathbf{g}(\eta))^2 - f(\eta)\mathbf{g'}(\eta),\tag{9}$$

subject to the same boundary conditions given in Equation (5). In [20], the authors applied the homotopy analysis method to solve the set of boundary value problems (1)–(6) in the absence of the second slip parameter (i.e., when *μ* = 0). However, Equation (1) with the boundary conditions (3) can be exactly solved, even in the presence of the second slip parameter *μ*, as will be introduced in the next section. This exact solution for *f*(*η*) will be then compared with the results obtained by [20] at a special case. Further, this exact formula for *f*(*η*) is to be inserted into Equation (9) to form with the boundary conditions (5) a single nonlinear differential equation in the unknown *g*(*η*). Details of the suggested procedure are presented in the next section.
